For any three vectors `veca, vecb, vecc` the value of `vecaxx(vecbxxvecc)+vecbxx(veccxxveca)+veccxx(vecaxxvecb)`, is

Options

A. `vec0`
B. `[(veca, vecb, vecc)]veca`
C. `[(veca, vecb, vecc)]vecb`
D. `[(veca, vecb, vecc)]vecc`

Explanation

To solve the problem, we need to evaluate the expression:

\[ \vec{a} \times (\vec{b} \times \vec{c}) + \vec{b} \times (\vec{c} \times \vec{a}) + \vec{c} \times (\vec{a} \times \vec{b}) \]

We can use the vector triple product identity, which states:

\[ \vec{x} \times (\vec{y} \times \vec{z}) = (\vec{x} \cdot \vec{z}) \vec{y} - (\vec{x} \cdot \vec{y}) \vec{z} \]

### Step-by-step Solution:

1. **Apply the Vector Triple Product Identity**:
- For the first term \(\vec{a} \times (\vec{b} \times \vec{c})\):
\[ \vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} \]

2. **Apply the Vector Triple Product Identity to the second term**:
- For the second term \(\vec{b} \times (\vec{c} \times \vec{a})\):
\[ \vec{b} \times (\vec{c} \times \vec{a}) = (\vec{b} \cdot \vec{a}) \vec{c} - (\vec{b} \cdot \vec{c}) \vec{a} \]

3. **Apply the Vector Triple Product Identity to the third term**:
- For the third term \(\vec{c} \times (\vec{a} \times \vec{b})\):
\[ \vec{c} \times (\vec{a} \times \vec{b}) = (\vec{c} \cdot \vec{b}) \vec{a} - (\vec{c} \cdot \vec{a}) \vec{b} \]

4. **Combine all three results**:
- Now, substituting back into the original expression:
\[ \begin{align*} & \left[(\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c}\right] + \left[(\vec{b} \cdot \vec{a}) \vec{c} - (\vec{b} \cdot \vec{c}) \vec{a}\right] + \left[(\vec{c} \cdot \vec{b}) \vec{a} - (\vec{c} \cdot \vec{a}) \vec{b}\right] \\ & = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} + (\vec{b} \cdot \vec{a}) \vec{c} - (\vec{b} \cdot \vec{c}) \vec{a} + (\vec{c} \cdot \vec{b}) \vec{a} - (\vec{c} \cdot \vec{a}) \vec{b} \end{align*} \]

5. **Simplify the expression**:
- Notice that \((\vec{a} \cdot \vec{b}) \vec{c}\) and \((\vec{b} \cdot \vec{a}) \vec{c}\) cancel each other out, as do \((\vec{c} \cdot \vec{a}) \vec{b}\) and \((\vec{a} \cdot \vec{c}) \vec{b}\), leading to:
\[ 0 \]

### Final Result:
Thus, the value of the expression is:

\[ \boxed{0} \]